Put the flour, beaten eggs, and panko in three separate shallow bowls for dredging.
Season both sides of each butterflied, pounded chicken breast with salt and pepper.
Dredge each chicken cutlet in flour, shaking off excess, dip into the beaten eggs, then press into the panko so the crumbs adhere evenly.
Heat 1/4 cup canola oil in each of two large skillets over moderately high heat until shimmering.
Add the chicken to the hot skillets and cook until golden and crispy, about 3 minutes per side, turning once.
Transfer cooked chicken to a paper towel–lined baking sheet and sprinkle lightly with salt.
While the chicken rests, melt the butter in a small saucepan over moderately high heat and cook until it browns and smells nutty, about 4 minutes, stirring so it does not burn.
Remove the saucepan from the heat and stir in the capers, lemon juice, and chopped parsley.
Spoon the browned-butter caper sauce over the schnitzels and serve immediately.